A001661 Largest number not the sum of distinct positive n-th powers. (Formerly M5393 N2342)

128, 12758, 5134240, 67898771, 11146309947, 766834015734, 4968618780985762

Offset

2,1

Comments

a(8) > 74^8. - Donovan Johnson, Nov 23 2010

Fuller and Nichols prove that a(6) = 11146309947 and that 2037573096 positive numbers cannot be written as the sum of distinct 6th powers. - Robert Nichols, Sep 09 2017

a(8) >= 83^8 ~ 2.25e15 since A030052(8) = 84. Similarly, a(9..15) >= (46^9, 62^10, 67^11, 80^12, 101^13, 94^14, 103^15) ~ (9.2e14, 8.4e17, 1.2e20, 6.9e22, 1.1e26, 4.2e27, 1.6e30), cf. formula. Most often a(n) will be closer to and even larger than A030052(n)^n. - In the literature, a(n)+1 is known as the anti-Waring number N(n,1). - M. F. Hasler, May 15 2020

a(9..16) > (1.55e17, 1.31e19, 1.64e21, 5.55e23, 1.32e26, 1.37e28, 2.09e30, 9.99e35). - Michael J. Wiener, Jun 10 2023

References

S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.

Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=2..8.

R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.

Shalosh B. Ekhad and Doron Zeilberger, Automating John P. D'Angelo's method to study Complete Polynomial Sequences, arXiv:2111.02832 [math.NT], 2021.

Mauro Fiorentini, Rappresentazione di interi come somma di potenze (in Italian).

C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.

R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285.

D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, arXiv:1610.02439 [math.NT], 2016-2017.

D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, J. Int. Seq. 20 (2017), #17.7.5.

P. LeVan and D. Prier, Improved Bounds on the Anti-Waring Number, J. Int. Seq. 20 (2017), #17.8.7.

D. C. Mayer, Sharp bounds for the partition function of integer sequences, BIT 27 (1987), 98-110.

D. C. Mayer, Partition functions via bit list operations, 2009.

N. J. A. Sloane and R. E. Dressler, Correspondence, June 1974

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.

Eric Weisstein's World of Mathematics, Waring's Problem

M. J. Wiener, The Largest Integer Not the Sum of Distinct 8th Powers, J. Integer Sequences, 26 (2023), Article 23.5.4.

J. W. Wrench, Jr., Letter to N. J. A. Sloane, 10 Apr, 1974

Formula

a(n) < d*2^(n-1)*(c*2^n + (2/3)*d*(4^n - 1) + 2*d - 2)^n + c*d, where c = n!*2^(n^2) and d = 2^(n^2 + 2*n)*c^(n-1) - 1, according to Kim [2016-2017]. - Danny Rorabaugh, Oct 11 2016

a(n) >= (A030052(n)-1)^n. - M. F. Hasler, May 15 2020

Crossrefs

Cf. A030052, A173563, A279529.

Cf. A121571 (primes instead of integers).

Sequence in context: A282398 A297101 A181250 * A264095 A239190 A283856

Adjacent sequences: A001658 A001659 A001660 * A001662 A001663 A001664

Keyword

nonn,nice,more,hard

Author

N. J. A. Sloane and Robert G. Wilson v

Extensions

a(7) from Donovan Johnson, Nov 23 2010

a(8) from Michael J. Wiener, Jun 10 2023

Status

approved